Let A and B be two unitary Banach algebras. We study linear mappings from A into B which preserve the polynomially convex hull of the spectrum. In particular, we give conditions under which such surjective linear mappings are Jordan morphisms.
We prove that the separating space of a Lie homomorphism from a Banach algebra onto a Banach algebra is contained in the centre modulo the radical.
A classical Julia-Carathéodory theorem concerning radial limits of holomorphic maps in one dimension is extended to hyperbolic contractions of bounded symmetric domains in J*-algebras.
The questions when a derivation on a Jordan-Banach algebra has quasi-nilpotent values, and when it has the range in the radical, are discussed.
We prove that, if A is an associative algebra with two commuting involutions τ and π, if A is a τ-π-tight envelope of the Jordan Triple System T:=H(A,τ) ∩ S(A,π), and if T is nondegenerate, then every complete norm on T making the triple product continuous is equivalent to the restriction to T of an algebra norm on A.
A generalization of a result of Cohen-Hewitt is given in the case of Jordan-Banach algebras. Some precisions of factorization are obtained.
Using an appropriate definition of the multiplicity of a spectral value, we introduce a new definition of the trace and determinant of elements with finite spectrum in Jordan-Banach algebras. We first extend a result obtained by J. Zemánek in the associative case, on the connectedness of projections which are close to each other spectrally (Theorem 2.3). Secondly we show that the rank of the Riesz projection associated to a finite-rank element a and a finite subset of its spectrum is equal to the...